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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Surfaces expanding by the power of the Gauss curvature flow

Author: Qi-Rui Li
Journal: Proc. Amer. Math. Soc. 138 (2010), 4089-4102
MSC (2010): Primary 53C44, 35K55
Published electronically: June 16, 2010
MathSciNet review: 2679630
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Abstract: In this paper, we describe the flow of 2-surfaces in $\mathbb {R}^{3}$ for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity $K^{-\alpha }$, when $\frac {1}{2}<\alpha \leq 1$, converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for $\alpha \in (0,\frac {1}{2}]$ by Urbas and Huisken and for $\alpha =1$ by Schnürer.

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Qi-Rui Li
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Keywords: Surface, expanding curvature flow, velocity function $K^{-\alpha }$
Received by editor(s): June 2, 2009
Received by editor(s) in revised form: October 9, 2009, November 26, 2009, December 19, 2009, December 31, 2009, and February 6, 2010
Published electronically: June 16, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.